For an ordinal probit model, \(\theta = (\beta , \gamma )\)  and \(f(y|z,\beta ,\gamma ) \equiv f(y|z,\gamma )\) ; whereas for a binary probit model \(\theta =\beta \)  and \(f(y|z,\beta ) \equiv f(y|z)\) . The two equivalencies arise because,
given a latent observation \(z_{i}\) , \(y_{i}\)  is known with certainty
regardless of \(\beta \)  for \(i=1,\ldots ,n\) . This de-linking of the likelihood
function from \(\beta \) , made possible through data augmentation, simplifies
the estimation procedure and allows sampling of \(\beta \)  through a Gibbs
process [1]} – a well known Markov chain Monte Carlo
(MCMC) technique. The latent variable \(z\)  is sampled element-wise from a
truncated normal distribution. For ordinal models, a monotone transformation
of the cut-points, \(\gamma \) , is sampled using an MH algorithm. The MCMC
algorithms for estimating ordinal and binary probit models outlined
here were introduced in [2]}. Other notable references
that describe the Bayesian modeling and estimation of ordinal and binary
responses in great detail include [3]},
[4]}, and [5]}. Bayesian
estimation of logit model is based on the same principle and presented in
[6]} and [5]}.
